Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Implications from propositions A => B, A <=> C and C => B

  1. Oct 6, 2007 #1
    I'm not 100% sure what this is in English so I'll try to describe it. Gives that:

    A: x^2 < 16
    B: -4 < x
    C: -4 < x < 4

    I'm supposed to put out every possibility for => and <=> between A,B and C. The key says that A => B, A <=> C and C => B. I can understand this, but isn't it true for every proposition (I think that it's called this) that A <=> A. That is, every proposition implies itself?
  2. jcsd
  3. Oct 6, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    Suppose you have three propositions, A, B and C and want to prove that they are equivalent. That means that
    A <=> B, B <=> C and A <=> C
    which can also be written as
    A => B, B => A, A => C, C => A, B => C, C => B.
    In words: if you know that one of them is true/false, they must all be true/false.
    Obviously, A <=> A is always true and it's not included in the list.

    You can prove all 6 of them consecutively, but that would be a lot of work. Therefore we find a shortcut:
    suppose you would be able to prove half of them, namely that A => B, B => C and C => A.
    Then because B implies C and C implies A, B also implies A so you automatically get B => A and therefore A <=> B.
    Similarly, C => A and A => B so C also implies B (via A) hence B <=> C.
    So proving these three statements will prove all six of them.

    This is what the key suggests. It is wrong though, in claiming that A => B, A <=> C and C => B suffices. For example, you cannot get B => A (there is no premise starting with B).
  4. Oct 13, 2007 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    I think you misunderstood the question. Nothing was said about A<=>B, B<=>C, C<=>A. there is nothing here that implies B=> A and the answer key does NOT suggest such a thing.
  5. Oct 14, 2007 #4


    User Avatar
    Science Advisor
    Homework Helper

    Probably then I got confused by the formulation.
    What is meant by "put out every possbility"?
    And I don't think I see the relevance of the question in the first part of the post to the question "isn't it true for every proposition ...".

    So I apologize if I mislead you x-is-y, perhaps you can try to rephrase the question (or if someone else understands it, explain it to my numb mind)?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Implications from propositions A => B, A <=> C and C => B
  1. A->b, C->~b, A /\ C->? (Replies: 3)